In the The Flash episode The Nuclear Man, we see Barry and Linda making out, and he vibrates. That brought the question to my mind. If Barry ever decides to have a kid, would it also have super speed because Barry's DNA has mutated and we tend to pass DNA to future generations?

2 Answers
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It depends on whether or not the mutation is "germ line". Each cell in your body has its own set of genes, and if a cell in your eye has a mutation, this won't be passed on to any children.

This is actually sort of a big deal, as medicine takes the first steps towards being able to cure genetic diseases. They have methods of altering the genes inside your cells such that if you had cystic fibrosis they might be able to force the cells of your lungs into behaving properly. However, medical ethicists are reluctant to approve any such therapy if there is the potential for the changes to be passed on to children. (Thankfully for cystic fibrosis sufferers, no one procreates with their lungs.)

Don Allen was the son of Barry and Iris Allen, conceived in the 30th
Century after his father's retirement, but born after his death in the
first Crisis. He and his twin sister Dawn inherited a fraction of his
speed

Dawn and her twin brother Don were born in the 30th century. Both
twins inherited the power of superspeed from their father, Barry
Allen. Despite widespread fear and hatred towards metahumans and their
powers in the 30th century, Dawn and Don chose to do good as
superheroes. Using their superspeed to spin around quickly, they
appeared as tornadoes and were known as the Tornado Twins.

Now if the tv show maintains these comics then we will see this genetic transfer in future.